Pre‐AP Physics: Unit 7Vibrations, Waves, and Sound

Clear Creek High School

Simple Harmonic Motion

Simple Harmonic Motion

– Constant periodic motion of an object.

– An object oscillates “back and forth” along the same path.

• Examples1. Mass on a string (pendulum)

2. Mass on a spring (spring oscillator)

Simple Harmonic Motion

– The restoring force is directly proportional to the displacement of the mass from equilibrium.– For springs, the restoring force

is determined by Hooke’s Law:F = -kx

– For pendulums, the restoring force is a component of the weight of the object.

Simple Harmonic Motion

– In a frictionless environment, the motion would continue on indefinitely.

– When friction is present, the motion slows until the object eventually comes to a stop.

– This is called Damping.

Simple Harmonic Motion

– SHM is similar to circular motion.

– Certain terms can be used to describe both.– Frequency

– # of oscillations (cycles) per second.

– Measured in Hertz

– Period

– The time needed to complete one oscillation.

– Measured in seconds.

Frequency and Period

• Frequency and period are inversely related:

Frequency and Period

• ExampleA swinging pendulum bob completes 4 oscillations every 2 seconds.1. What is the frequency of the

pendulum?Answer: 2 Hz

2. What is the period of oscillation?Answer: 0.5 seconds

Calculating Period

• Period of oscillation for a pendulum.– Only dependent on the length of the pendulum (and gravity).

– Equation:• l: length of pendulum (in meters)

• g: acceleration due to gravity (9.8 m/s2)

• Period of oscillation for an object attached to a spring.– Only dependent on the mass of the object (and the spring

constant of the spring).

– Equation:• m: mass of the object (in kg)

• k: spring constant of the spring (in N/m)

Example 1A 1.5 kg object is attached to a fixed spring and placed on a horizontal surface that is frictionless. The spring constant is found to be 25 N/m. The object is pulled to a maximum amplitude of 0.25 meters and released. What is the period of the oscillation of the object?

Answer: 1.54 seconds

Example 2A clockmaker is trying to produce a large grandfather clock for a customer. He would like it to contain of a pendulum that will move back and forth once every second. How long should the clockmaker design the pendulum rod for this clock?

Answer: 0.248 meters

Wave Properties

What is a Wave?

– A disturbance that carries energy as it propagates through matter or space.

What is a Wave?

The Two Major Categories of Waves• Mechanical Waves

– Require a physical medium to travel through.

o Examples– Sound waves

– Water waves

– Spring waves

The Two Major Categories of Waves• Electromagnetic Waves

– Do not require a physical medium.

o Example: – light can travel through empty space

Types of Waves

• Transverse Waves– Particles move perpendicular to the wave.

Types of Waves

• Longitudinal Waves– Particles move parallel to the wave.

Types of Waves

• All electromagnetic waves are transverse and mechanical waves can be transverse or longitudinal.

Wave Description• Transverse Waves

o Crest

o Trough

o Amplitude– Maximum displacement from equilibrium position.

– How much energy the wave is carrying.

o Wavelength– The distance from one point on a wave to the same point on the next

wave.

Wave Description• Longitudinal Waves

o Compression– Region of higher

concentration.

o Rarefaction– Region of lower concentration.

o Intensity– How much energy the wave is carrying.

o Wavelength– The distance from one compression/rarefaction to the next.

Wave Description• Frequency and Period of Waves

– Same as with simple harmonic motion.

– Frequency• Number of waves that pass by

each second.

• Measured in Hertz.

– Period• The time needed for each wave

to pass by.

• Measured in seconds.

Wave Description• Wave Pulses vs. Periodic Waves

o Wave Pulse– a single, independent

wave.

o Periodic Wave– produced by a continuously oscillating source.

Properties of Waves• Wave Motion

– Waves transfer energy!

– Matter has little or no net movement.

Properties of Waves

• Wave Speedo Equation

v : wave velocity (in m/s)f : frequency (in Hertz)λ : wavelength (in meters)

λ = “Lambda”

Example 3A person standing in a canyon plays a note from a trumpet with a frequency of 110 Hz. The speed of sound at that spot is 330 m/s.a. What is the wavelength of the sound wave?– Answer: 3 meters

b. The sound wave reflects off of a canyon wall and the person hears the echo 4 seconds later. How far away is the person from the wall?– Answer: 660 meters

Properties of Waves

• Reflection– When a wave encounters a boundary and “bounces” off.

Properties of Waves

• Reflectiono Reflection from a free end

– The wave returns on the same side (in phase).

Properties of Waves

• Reflectiono Reflection from a fixed end

– The wave returns on the opposite side (out of phase).

Properties of Waves

• Refraction– The bending of a wave as it enters a new medium.

– This happens because the wave changes speed.

Properties of Waves

• Diffraction– The bending of a wave around the edge of a boundary or

obstacle.

Sound Waves

Sound Waves

• A mechanical longitudinal wave that travels through a compressible medium (such as air or water).

• Sound (like other waves) is formed by the vibration of an object as it compresses the air around it.– Examples:

• Vocal chords

• Speakers

• Stringed Instruments

Frequency of Sound

• The frequency of a sound wave determines the pitch that is heard.

• Human ears can hear frequencies in the range of 20 Hz – 20,000 Hz– Infrasonic – frequencies below 20 Hz

– Ultrasonic – frequencies above 20,000 Hz

Speed of Sound

• Dependent on the medium through which the sound wave travels.

• Denser medium = faster

• Higher temperature = faster

Sound Intensity– How “loud” a sound is.

– Sound waves propagate out in 3-dimensions from their source.

– Intensity is the rate at which energy is transferred through a certain area.

– Equation

– I: Intensity in watts per meters squared (W/m2)

– P : power output of the source (in Watts)

– r : distance from the source (in meters)

Example 4What is the intensity of the sound waves produced by a trombone at a distance of 3.2 meters when the power output of the trombone is 0.20 Watts?

Answer: 0.00155 W/m2

Decibels• Another way to measure wave intensity (sound volume).

• Decibels represent the relative intensity of the wave to a certain reference intensity.

• Equation

– I : Intensity at a certain point (in W/m2)

– I0 : reference intensity (1.0 x 10-12 W/m2, threshold of human hearing)

• Decibels are dimensionless numbers; they are just a ratio of intensities.

• Each 10dB increase corresponds to an increase in intensity by a factor of 10.

Example 5Suzie finds that the intensity of the sound waves where she is located at a rock concert is 0.316 W/m2. What is the decibel level of these sound waves?

Answer: 115 dB

Example 6A very loud lawnmower is producing sound waves with a power output of 0.40 Watts. What is the relative intensity (in decibels) of the sound waves 5 meters from the lawnmower?

Answer: 91 dB

The Doppler Effect• A change in the observed frequency of a wave caused by relative

motion between the source and observer.

• Moving closer = higher frequency

• Moving away = lower frequency

The Doppler Effect

• Shock Wave– Created by an object moving faster than the speed of a wave

it is producing.

– With sound waves this can create a sonic boom.

Wave Interactions

Wave Interactions– When two (or more) waves are travelling through the same

medium they will interact with each other.

• The Principle of Superposition – The net displacement is the sum of the individual wave displacements.

Wave Interference

• Constructive Interference– The waves are in-phase.

– The amplitudes add together.

Wave Interference

• Destructive Interference– The waves are out-of-phase.

– The amplitudes “cancel out”

Example 7Shown to the right are four different pairs of wave pulses that move toward and interact with each other. Rank, from most to least, the height of the peak that results when the two wave pulses coincide.

Answer: A,B,D,C

Interference in Sound Waves

• Sound waves of slightly different frequencies can interfere with one another and cause beats.

• The number of beats per second corresponds to the difference between frequencies.

Standing Waves• A repeated wave that is continuously reflected produces a

pattern caused by the rotation of constructive and destructive interference.

Standing Waves

• Nodes– The point(s) where

displacement is zero.

• Antinodes– The point(s) of maximum

displacement.

Standing Waves

• Natural (Fundamental) Frequency– The lowest frequency at which an

object will vibrate to produce a standing wave with one antinode.

• Harmonics– Multiples of the natural

frequency of a wave.

• Frequency and wavelength are inversely related:v = f

Example 8A musician plucks a guitar string and it begins to vibrate. The string has a total length of 0.75 meters.

a) What is the wavelength of the first harmonic?

b) What is the wavelength of the third harmonic?

c) What is the frequency of the sixth harmonic if thespeed of the waves is 338 m/s?

Answers: 1.5m, 0.5 m, 1352 Hz

Resonance

Resonance

• Resonance occurs when an object is made to vibrate at its natural frequency.

• The vibration of the object forms a standing wave.

• Can result in a large increase in amplitude.

Resonance Tubes• If air moves quickly across the opening of a tube, a sound is

produced.

• The air begins to resonate in the tube, forming a standing wave.

• These standing waves can have harmonicsand can occur in tubes with one open end or with two open ends.

• The opening of a tube will always contain an antinode and a barrier will always contain a node.

1st 2nd 3rd 4th

Example 9Jeffrey blows air across the opening of a 15 cm bottle and it makes a sound at its fundamental frequency.

a) What is the wavelength of the sound wave produced in the bottle?

b) What is the fundamental frequency? The speed of sound in the air inside the bottle is 337 m/s.

Answers: 0.6m, 562 Hz

Example 10An organ pipe is 2.5 meters long and is open on both ends. The speed of sound in the air around and inside the pipe is 341 m/s.

a) What is the fundamental frequency of a sound wave produced in this pipe?

b) What is the wavelength of the 3rd harmonic?

c) What is the frequency of the 6th harmonic?Answers: 68.2 Hz, 1.67 meters, 409.2 Hz